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NEAR-WALL ENSTROPHY GENERATION IN A DRAG-REDUCED

TURBULENT CHANNEL FLOW WITH SPANWISE WALL OSCILLATIONS

Pierre Ricco1, Claudio Ottonelli, Yosuke Hasegawa, Maurizio Quadrio

1 Sheffield, 2 Onera Paris, 3 Tokyo, 4 Politecnico Milano

ERCOFTAC/PLASMAERO Workshop, Toulouse, 10 December 2012

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

1-28

TURBULENT DRAG REDUCTION

ACTIVE OPEN-LOOP TECHNIQUE Energy input into system Pre-determined forcing NUMERICAL APPROACH Direct numerical simulations of wall turbulence Fully-developed turbulent channel flow (Reτ = uτh/ν = 200) Compact finite-difference scheme along wall-normal direction Spectral discretization along streamwise and spanwise directions SPANWISE WALL OSCILLATIONS New approach: Turbulent enstrophy Transient evolution CONSTANT DP/DX τw is fixed in fully-developed conditions GAIN: Ub increases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

2-28

TURBULENT DRAG REDUCTION

ACTIVE OPEN-LOOP TECHNIQUE Energy input into system Pre-determined forcing NUMERICAL APPROACH Direct numerical simulations of wall turbulence Fully-developed turbulent channel flow (Reτ = uτh/ν = 200) Compact finite-difference scheme along wall-normal direction Spectral discretization along streamwise and spanwise directions SPANWISE WALL OSCILLATIONS New approach: Turbulent enstrophy Transient evolution CONSTANT DP/DX τw is fixed in fully-developed conditions GAIN: Ub increases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

2-28

TURBULENT DRAG REDUCTION

ACTIVE OPEN-LOOP TECHNIQUE Energy input into system Pre-determined forcing NUMERICAL APPROACH Direct numerical simulations of wall turbulence Fully-developed turbulent channel flow (Reτ = uτh/ν = 200) Compact finite-difference scheme along wall-normal direction Spectral discretization along streamwise and spanwise directions SPANWISE WALL OSCILLATIONS New approach: Turbulent enstrophy Transient evolution CONSTANT DP/DX τw is fixed in fully-developed conditions GAIN: Ub increases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

2-28

TURBULENT DRAG REDUCTION

ACTIVE OPEN-LOOP TECHNIQUE Energy input into system Pre-determined forcing NUMERICAL APPROACH Direct numerical simulations of wall turbulence Fully-developed turbulent channel flow (Reτ = uτh/ν = 200) Compact finite-difference scheme along wall-normal direction Spectral discretization along streamwise and spanwise directions SPANWISE WALL OSCILLATIONS New approach: Turbulent enstrophy Transient evolution CONSTANT DP/DX τw is fixed in fully-developed conditions GAIN: Ub increases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

2-28

TURBULENT DRAG REDUCTION

ACTIVE OPEN-LOOP TECHNIQUE Energy input into system Pre-determined forcing NUMERICAL APPROACH Direct numerical simulations of wall turbulence Fully-developed turbulent channel flow (Reτ = uτh/ν = 200) Compact finite-difference scheme along wall-normal direction Spectral discretization along streamwise and spanwise directions SPANWISE WALL OSCILLATIONS New approach: Turbulent enstrophy Transient evolution CONSTANT DP/DX τw is fixed in fully-developed conditions GAIN: Ub increases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

2-28

SPANWISE WALL OSCILLATIONS

GEOMETRY

Mean flow x y z Lx Ly Lz Ww = A sin 2π T t

R =

Cf,r −Cf,o Cf,r

=

U2

b,o−U2 b,r

U2

b,o

Why does the skin-friction coefficent decrease? Cf = 2τw/(ρU2

b) decreases → study why Ub increases 5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

3-28

SPANWISE WALL OSCILLATIONS

GEOMETRY

Mean flow x y z Lx Ly Lz Ww = A sin 2π T t

R =

Cf,r −Cf,o Cf,r

=

U2

b,o−U2 b,r

U2

b,o

Why does the skin-friction coefficent decrease? Cf = 2τw/(ρU2

b) decreases → study why Ub increases 5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

3-28

SPANWISE WALL OSCILLATIONS

GEOMETRY

Mean flow x y z Lx Ly Lz Ww = A sin 2π T t

R =

Cf,r −Cf,o Cf,r

=

U2

b,o−U2 b,r

U2

b,o

Why does the skin-friction coefficent decrease? Cf = 2τw/(ρU2

b) decreases → study why Ub increases 5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

3-28

SPANWISE WALL OSCILLATIONS

GEOMETRY

Mean flow x y z Lx Ly Lz Ww = A sin 2π T t

R =

Cf,r −Cf,o Cf,r

=

U2

b,o−U2 b,r

U2

b,o

Why does the skin-friction coefficent decrease? Cf = 2τw/(ρU2

b) decreases → study why Ub increases 5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

3-28

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f(y, t) = 1 LxLz

Lx Lz

f(x, y, z, t)dzdx

PHASE

• f(y, τ) = 1

N

N−1

• n=0

f(y, nT + τ)

TIME

f (y) = 1 T

T

f(y, τ)dτ

GLOBAL

[f]g =

h

f (y)dy

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

4-28

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f(y, t) = 1 LxLz

Lx Lz

f(x, y, z, t)dzdx

PHASE

• f(y, τ) = 1

N

N−1

• n=0

f(y, nT + τ)

TIME

f (y) = 1 T

T

f(y, τ)dτ

GLOBAL

[f]g =

h

f (y)dy

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

4-28

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f(y, t) = 1 LxLz

Lx Lz

f(x, y, z, t)dzdx

PHASE

• f(y, τ) = 1

N

N−1

• n=0

f(y, nT + τ)

TIME

f (y) = 1 T

T

f(y, τ)dτ

GLOBAL

[f]g =

h

f (y)dy

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

4-28

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f(y, t) = 1 LxLz

Lx Lz

f(x, y, z, t)dzdx

PHASE

• f(y, τ) = 1

N

N−1

• n=0

f(y, nT + τ)

TIME

f (y) = 1 T

T

f(y, τ)dτ

GLOBAL

[f]g =

h

f (y)dy

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

4-28

AVERAGING OPERATORS

SPACE: HOMOGENEOUS DIRECTIONS

f(y, t) = 1 LxLz

Lx Lz

f(x, y, z, t)dzdx

PHASE

• f(y, τ) = 1

N

N−1

• n=0

f(y, nT + τ)

TIME

f (y) = 1 T

T

f(y, τ)dτ

GLOBAL

[f]g =

h

f (y)dy

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

4-28

MEAN FLOW

25 50 75 100 125 150 175 0.05 0.1 0.15 0.2 0.25 0.3 0.35 10 10

1

10

2

5 10 15 20 25

y

• U

R T

Scaling by viscous units Mean velocity increases in the bulk of the channel Mean wall-shear stress is unchanged Optimum period of oscillation T≈75

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

5-28

MEAN FLOW

25 50 75 100 125 150 175 0.05 0.1 0.15 0.2 0.25 0.3 0.35 10 10

1

10

2

5 10 15 20 25

y

• U

R T

Scaling by viscous units Mean velocity increases in the bulk of the channel Mean wall-shear stress is unchanged Optimum period of oscillation T≈75

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

5-28

MEAN FLOW

25 50 75 100 125 150 175 0.05 0.1 0.15 0.2 0.25 0.3 0.35 10 10

1

10

2

5 10 15 20 25

y

• U

R T

Scaling by viscous units Mean velocity increases in the bulk of the channel Mean wall-shear stress is unchanged Optimum period of oscillation T≈75

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

5-28

MEAN FLOW

25 50 75 100 125 150 175 0.05 0.1 0.15 0.2 0.25 0.3 0.35 10 10

1

10

2

5 10 15 20 25

y

• U

R T

Scaling by viscous units Mean velocity increases in the bulk of the channel Mean wall-shear stress is unchanged Optimum period of oscillation T≈75

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

5-28

MEAN FLOW

25 50 75 100 125 150 175 0.05 0.1 0.15 0.2 0.25 0.3 0.35 10 10

1

10

2

5 10 15 20 25

y

• U

R T

Scaling by viscous units Mean velocity increases in the bulk of the channel Mean wall-shear stress is unchanged Optimum period of oscillation T≈75

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

5-28

TURBULENCE STATISTICS

10 10

1

10

2

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 10 10

1

10

2

• 1

1 2 3 4 5 6 7

y

• u2
• ,
• v2
• ,
• w2
• ,

uv y

• vw

φ=0 φ= π 4 φ= π 2 φ= 3π 4

Turbulence kinetic energy decreases Streamwise velocity fluctuations are attenuated the most New oscillatory Reynolds stress term vw in created, vw = 0

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

6-28

TURBULENCE STATISTICS

10 10

1

10

2

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 10 10

1

10

2

• 1

1 2 3 4 5 6 7

y

• u2
• ,
• v2
• ,
• w2
• ,

uv y

• vw

φ=0 φ= π 4 φ= π 2 φ= 3π 4

Turbulence kinetic energy decreases Streamwise velocity fluctuations are attenuated the most New oscillatory Reynolds stress term vw in created, vw = 0

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

6-28

TURBULENCE STATISTICS

10 10

1

10

2

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 10 10

1

10

2

• 1

1 2 3 4 5 6 7

y

• u2
• ,
• v2
• ,
• w2
• ,

uv y

• vw

φ=0 φ= π 4 φ= π 2 φ= 3π 4

Turbulence kinetic energy decreases Streamwise velocity fluctuations are attenuated the most New oscillatory Reynolds stress term vw in created, vw = 0

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

6-28

TURBULENCE STATISTICS

10 10

1

10

2

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 10 10

1

10

2

• 1

1 2 3 4 5 6 7

y

• u2
• ,
• v2
• ,
• w2
• ,

uv y

• vw

φ=0 φ= π 4 φ= π 2 φ= 3π 4

Turbulence kinetic energy decreases Streamwise velocity fluctuations are attenuated the most New oscillatory Reynolds stress term vw in created, vw = 0

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

6-28

ENERGY BALANCE: A SCHEMATIC

DU DW DT Puv Pvw Ubτw Ew MKE-x MKE-z TKE

+3.5

15.9

+13.2 +12.9 +2.7

9.4

+0.8

6.5

+0.3 +1.1 6.5

Energy is fed through Px (→ Ubτw) and wall motion (→ Ew) Energy is dissipated through:

Mean-flow viscous effects (→ DU, DW ) Turbulent viscous effects (→ DT )

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

7-28

ENERGY BALANCE: A SCHEMATIC

DU DW DT Puv Pvw Ubτw Ew MKE-x MKE-z TKE

+3.5

15.9

+13.2 +12.9 +2.7

9.4

+0.8

6.5

+0.3 +1.1 6.5

Energy is fed through Px (→ Ubτw) and wall motion (→ Ew) Energy is dissipated through:

Mean-flow viscous effects (→ DU, DW ) Turbulent viscous effects (→ DT )

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

7-28

ENERGY BALANCE: A SCHEMATIC

DU DW DT Puv Pvw Ubτw Ew MKE-x MKE-z TKE

+3.5

15.9

+13.2 +12.9 +2.7

9.4

+0.8

6.5

+0.3 +1.1 6.5

Energy is fed through Px (→ Ubτw) and wall motion (→ Ew) Energy is dissipated through:

Mean-flow viscous effects (→ DU, DW ) Turbulent viscous effects (→ DT )

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

7-28

ENERGY BALANCE: A SCHEMATIC

DU DW DT Puv Pvw Ubτw Ew MKE-x MKE-z TKE

+3.5

15.9

+13.2 +12.9 +2.7

9.4

+0.8

6.5

+0.3 +1.1 6.5

Energy is fed through Px (→ Ubτw) and wall motion (→ Ew) Energy is dissipated through:

Mean-flow viscous effects (→ DU, DW ) Turbulent viscous effects (→ DT )

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

7-28

ENERGY BALANCE: EQUATIONS

GLOBAL MEAN KINETIC ENERGY EQUATION

Ubτw +

• A ∂

W ∂y

• y=0
• Ew

= −

• uv ∂

U ∂y

• g
• Puv

−

• vw ∂

W ∂y

• g
• Pvw

+

• ∂

U ∂y

2

g

• DU

+

• ∂

W ∂y

2

g

• DW

GLOBAL TURBULENT KINETIC ENERGY EQUATION

• uv ∂

U ∂y

• g
• Puv

+

• vw ∂

W ∂y

• g
• Pvw

+ ∂ui ∂xj ∂ui ∂xj

• g

= 0

TOTAL KINETIC ENERGY BALANCE

Ubτw + Ew = DU + DW + DT

TURBULENT DISSIPATION

DT = ωiωi

• g

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

8-28

ENERGY BALANCE: EQUATIONS

GLOBAL MEAN KINETIC ENERGY EQUATION

Ubτw +

• A ∂

W ∂y

• y=0
• Ew

= −

• uv ∂

U ∂y

• g
• Puv

−

• vw ∂

W ∂y

• g
• Pvw

+

• ∂

U ∂y

2

g

• DU

+

• ∂

W ∂y

2

g

• DW

GLOBAL TURBULENT KINETIC ENERGY EQUATION

• uv ∂

U ∂y

• g
• Puv

+

• vw ∂

W ∂y

• g
• Pvw

+ ∂ui ∂xj ∂ui ∂xj

• g

= 0

TOTAL KINETIC ENERGY BALANCE

Ubτw + Ew = DU + DW + DT

TURBULENT DISSIPATION

DT = ωiωi

• g

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

8-28

ENERGY BALANCE: EQUATIONS

GLOBAL MEAN KINETIC ENERGY EQUATION

Ubτw +

• A ∂

W ∂y

• y=0
• Ew

= −

• uv ∂

U ∂y

• g
• Puv

−

• vw ∂

W ∂y

• g
• Pvw

+

• ∂

U ∂y

2

g

• DU

+

• ∂

W ∂y

2

g

• DW

GLOBAL TURBULENT KINETIC ENERGY EQUATION

• uv ∂

U ∂y

• g
• Puv

+

• vw ∂

W ∂y

• g
• Pvw

+ ∂ui ∂xj ∂ui ∂xj

• g

= 0

TOTAL KINETIC ENERGY BALANCE

Ubτw + Ew = DU + DW + DT

TURBULENT DISSIPATION

DT = ωiωi

• g

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

8-28

ENERGY BALANCE: EQUATIONS

GLOBAL MEAN KINETIC ENERGY EQUATION

Ubτw +

• A ∂

W ∂y

• y=0
• Ew

= −

• uv ∂

U ∂y

• g
• Puv

−

• vw ∂

W ∂y

• g
• Pvw

+

• ∂

U ∂y

2

g

• DU

+

• ∂

W ∂y

2

g

• DW

GLOBAL TURBULENT KINETIC ENERGY EQUATION

• uv ∂

U ∂y

• g
• Puv

+

• vw ∂

W ∂y

• g
• Pvw

+ ∂ui ∂xj ∂ui ∂xj

• g

= 0

TOTAL KINETIC ENERGY BALANCE

Ubτw + Ew = DU + DW + DT

TURBULENT DISSIPATION

DT = ωiωi

• g

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

8-28

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease? Why does Ub increase?

THREE POSSIBILITIES

1

Stokes layer acts on DU directly

→ excluded because W does not work directly on ∂ U/∂y2

2

Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3

Stokes layer acts on DT = ωiωi

• g directly

→ W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi The term enstrophy was coined by G. Nickel and is from Greek στρoφ´ η, which means turn

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

9-28

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease? Why does Ub increase?

THREE POSSIBILITIES

1

Stokes layer acts on DU directly

→ excluded because W does not work directly on ∂ U/∂y2

2

Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3

Stokes layer acts on DT = ωiωi

• g directly

→ W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi The term enstrophy was coined by G. Nickel and is from Greek στρoφ´ η, which means turn

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

9-28

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease? Why does Ub increase?

THREE POSSIBILITIES

1

Stokes layer acts on DU directly

→ excluded because W does not work directly on ∂ U/∂y2

2

Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3

Stokes layer acts on DT = ωiωi

• g directly

→ W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi The term enstrophy was coined by G. Nickel and is from Greek στρoφ´ η, which means turn

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

9-28

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease? Why does Ub increase?

THREE POSSIBILITIES

1

Stokes layer acts on DU directly

→ excluded because W does not work directly on ∂ U/∂y2

2

Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3

Stokes layer acts on DT = ωiωi

• g directly

→ W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi The term enstrophy was coined by G. Nickel and is from Greek στρoφ´ η, which means turn

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

9-28

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease? Why does Ub increase?

THREE POSSIBILITIES

1

Stokes layer acts on DU directly

→ excluded because W does not work directly on ∂ U/∂y2

2

Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3

Stokes layer acts on DT = ωiωi

• g directly

→ W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi The term enstrophy was coined by G. Nickel and is from Greek στρoφ´ η, which means turn

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

9-28

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease? Why does Ub increase?

THREE POSSIBILITIES

1

Stokes layer acts on DU directly

→ excluded because W does not work directly on ∂ U/∂y2

2

Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3

Stokes layer acts on DT = ωiωi

• g directly

→ W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi The term enstrophy was coined by G. Nickel and is from Greek στρoφ´ η, which means turn

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

9-28

KEY QUESTIONS

STILL TO BE ANSWERED

Why does TKE decrease? Why does Ub increase?

THREE POSSIBILITIES

1

Stokes layer acts on DU directly

→ excluded because W does not work directly on ∂ U/∂y2

2

Stokes layer acts on Puv directly

→ excluded because W does not work directly on uv

3

Stokes layer acts on DT = ωiωi

• g directly

→ W works on turbulent vorticity transport

TURBULENT ENSTROPHY TRANSPORT

Study the transport of turbulent enstrophy ωiωi The term enstrophy was coined by G. Nickel and is from Greek στρoφ´ η, which means turn

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

9-28

TURBULENT ENSTROPHY EQUATION

1 2 ∂ ωiωi ∂τ

1

= ωxωy ∂ U ∂y

2

+

• ωzωy

∂ W ∂y

• 3

+

• ωj

∂u ∂xj ∂ W ∂y

• 4

−

• ωj

∂w ∂xj ∂ U ∂y

• 5

− vωx ∂2 W ∂y2

• 6

+ vωz ∂2 U ∂y2

7

+

• ωiωj

∂ui ∂xj

8

− 1 2 ∂ ∂y

• vωiωi
• 9

+ 1 2 ∂2 ωiωi ∂y2

10

−

• ∂ωi

∂xj ∂ωi ∂xj

11

.

Stokes layer influences dynamics of turbulent enstrophy Three terms: which is the dominating one?

→ Let’s look at the terms of the equation

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

10-28

TURBULENT ENSTROPHY EQUATION

1 2 ∂ ωiωi ∂τ

1

= ωxωy ∂ U ∂y

2

+

• ωzωy

∂ W ∂y

• 3

+

• ωj

∂u ∂xj ∂ W ∂y

• 4

−

• ωj

∂w ∂xj ∂ U ∂y

• 5

− vωx ∂2 W ∂y2

• 6

+ vωz ∂2 U ∂y2

7

+

• ωiωj

∂ui ∂xj

8

− 1 2 ∂ ∂y

• vωiωi
• 9

+ 1 2 ∂2 ωiωi ∂y2

10

−

• ∂ωi

∂xj ∂ωi ∂xj

11

.

Stokes layer influences dynamics of turbulent enstrophy Three terms: which is the dominating one?

→ Let’s look at the terms of the equation

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

10-28

TURBULENT ENSTROPHY EQUATION

1 2 ∂ ωiωi ∂τ

1

= ωxωy ∂ U ∂y

2

+

• ωzωy

∂ W ∂y

• 3

+

• ωj

∂u ∂xj ∂ W ∂y

• 4

−

• ωj

∂w ∂xj ∂ U ∂y

• 5

− vωx ∂2 W ∂y2

• 6

+ vωz ∂2 U ∂y2

7

+

• ωiωj

∂ui ∂xj

8

− 1 2 ∂ ∂y

• vωiωi
• 9

+ 1 2 ∂2 ωiωi ∂y2

10

−

• ∂ωi

∂xj ∂ωi ∂xj

11

.

Stokes layer influences dynamics of turbulent enstrophy Three terms: which is the dominating one?

→ Let’s look at the terms of the equation

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

10-28

TURBULENT ENSTROPHY EQUATION

1 2 ∂ ωiωi ∂τ

1

= ωxωy ∂ U ∂y

2

+

• ωzωy

∂ W ∂y

• 3

+

• ωj

∂u ∂xj ∂ W ∂y

• 4

−

• ωj

∂w ∂xj ∂ U ∂y

• 5

− vωx ∂2 W ∂y2

• 6

+ vωz ∂2 U ∂y2

7

+

• ωiωj

∂ui ∂xj

8

− 1 2 ∂ ∂y

• vωiωi
• 9

+ 1 2 ∂2 ωiωi ∂y2

10

−

• ∂ωi

∂xj ∂ωi ∂xj

11

.

Stokes layer influences dynamics of turbulent enstrophy Three terms: which is the dominating one?

→ Let’s look at the terms of the equation

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

10-28

TURBULENT ENSTROPHY EQUATION

1 2 ∂ ωiωi ∂τ

1

= ωxωy ∂ U ∂y

2

+

• ωzωy

∂ W ∂y

• 3

+

• ωj

∂u ∂xj ∂ W ∂y

• 4

−

• ωj

∂w ∂xj ∂ U ∂y

• 5

− vωx ∂2 W ∂y2

• 6

+ vωz ∂2 U ∂y2

7

+

• ωiωj

∂ui ∂xj

8

− 1 2 ∂ ∂y

• vωiωi
• 9

+ 1 2 ∂2 ωiωi ∂y2

10

−

• ∂ωi

∂xj ∂ωi ∂xj

11

.

Stokes layer influences dynamics of turbulent enstrophy Three terms: which is the dominating one?

→ Let’s look at the terms of the equation

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

10-28

TURBULENT ENSTROPHY PROFILES

FIXED WALL

10 10

1

10

2

• 0.025
• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015

y 2 5 7 8 9 10 11

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

11-28

TURBULENT ENSTROPHY PROFILES

OSCILLATING-WALL PROFILES

10 10

1

10

2

• 0.025
• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015

y 2 3 4 5 6 7 8 9 10 11

Term 3, ωzωy∂ W/∂y → turbulent enstrophy production is dominant Other oscillating-wall terms are much smaller Turbulent dissipation of turbulent enstrophy increases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

12-28

TURBULENT ENSTROPHY PROFILES

OSCILLATING-WALL PROFILES

10 10

1

10

2

• 0.025
• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015

y 2 3 4 5 6 7 8 9 10 11

Term 3, ωzωy∂ W/∂y → turbulent enstrophy production is dominant Other oscillating-wall terms are much smaller Turbulent dissipation of turbulent enstrophy increases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

12-28

TURBULENT ENSTROPHY PROFILES

OSCILLATING-WALL PROFILES

10 10

1

10

2

• 0.025
• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015

y 2 3 4 5 6 7 8 9 10 11

Term 3, ωzωy∂ W/∂y → turbulent enstrophy production is dominant Other oscillating-wall terms are much smaller Turbulent dissipation of turbulent enstrophy increases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

12-28

TURBULENT ENSTROPHY PROFILES

OSCILLATING-WALL PROFILES

10 10

1

10

2

• 0.025
• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015

y 2 3 4 5 6 7 8 9 10 11

Term 3, ωzωy∂ W/∂y → turbulent enstrophy production is dominant Other oscillating-wall terms are much smaller Turbulent dissipation of turbulent enstrophy increases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

12-28

INTERESTING, BUT...

We have not answered questions on TKE and Ub, yet Key: transient from start-up of wall motion

100 200 300 400 500 2.5 5 7.5 10 12.5 15

t

USEFUL INFORMATION

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

13-28

INTERESTING, BUT...

We have not answered questions on TKE and Ub, yet Key: transient from start-up of wall motion

100 200 300 400 500 2.5 5 7.5 10 12.5 15

t Term 3

USEFUL INFORMATION

RED: term 3 increases abruptly, then decreases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

14-28

INTERESTING, BUT...

We have not answered questions on TKE and Ub, yet Key: transient from start-up of wall motion

100 200 300 400 500 2.5 5 7.5 10 12.5 15

t Term 3 Enstrophy

USEFUL INFORMATION

RED: term 3 increases abruptly, then decreases BLACK: turbulent enstrophy increases, then decreases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

15-28

INTERESTING, BUT...

We have not answered questions on TKE and Ub, yet Key: transient from start-up of wall motion

100 200 300 400 500 2.5 5 7.5 10 12.5 15

t Term 3 Enstrophy

USEFUL INFORMATION

RED: term 3 increases abruptly, then decreases BLACK: turbulent enstrophy increases, then decreases

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

16-28

INTERESTING, BUT...

We have not answered questions on TKE and Ub, yet Key: transient from start-up of wall motion

100 200 300 400 500 2.5 5 7.5 10 12.5 15

t Term 3 Enstrophy TKE

USEFUL INFORMATION

RED: term 3 increases abruptly, then decreases BLACK: turbulent enstrophy increases, then decreases BLUE: TKE decreases monotonically

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

17-28

TRANSIENT: THREE STAGES

SHORT STAGE

Turbulent enstrophy increases through ωzωy∂ W/∂y

INTERMEDIATE STAGE

TKE decreases because of enhanced turbulent dissipation

LONG STAGE

Bulk velocity increases because of TKE reduction → drag reduction

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

18-28

TRANSIENT: THREE STAGES

SHORT STAGE

Turbulent enstrophy increases through ωzωy∂ W/∂y

INTERMEDIATE STAGE

TKE decreases because of enhanced turbulent dissipation

LONG STAGE

Bulk velocity increases because of TKE reduction → drag reduction

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

18-28

TRANSIENT: THREE STAGES

SHORT STAGE

Turbulent enstrophy increases through ωzωy∂ W/∂y

INTERMEDIATE STAGE

TKE decreases because of enhanced turbulent dissipation

LONG STAGE

Bulk velocity increases because of TKE reduction → drag reduction

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

18-28

TRANSIENT: THREE STAGES

SHORT STAGE

Turbulent enstrophy increases through ωzωy∂ W/∂y

INTERMEDIATE STAGE

TKE decreases because of enhanced turbulent dissipation

LONG STAGE

Bulk velocity increases because of TKE reduction → drag reduction

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

18-28

DRAG REDUCTION MECHANISM

Initial state

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

19-28

DRAG REDUCTION MECHANISM

Short t < 50 Initial state ωzωy ∂W

∂y ↑

ωiωi ↑

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

20-28

DRAG REDUCTION MECHANISM

Short t < 50 Initial state ωzωy ∂W

∂y ↑

ωiωi ↑ DT ↑

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

21-28

DRAG REDUCTION MECHANISM

Short t < 50 Intermediate 50 < t < 400 Initial state

∂uv ∂y ↓

ωzωy ∂W

∂y ↑

ωiωi ↑ DT ↑ TKE ↓

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

22-28

DRAG REDUCTION MECHANISM

Short t < 50 Intermediate 50 < t < 400 Initial state ∂U ∂t > 0

∂uv ∂y ↓

ωzωy ∂W

∂y ↑

ωiωi ↑ DT ↑ TKE ↓

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

23-28

DRAG REDUCTION MECHANISM

Short t < 50 Intermediate 50 < t < 400 Long t > 400 Initial state ‘Drag reduction’

h

0 Udy ↑

∂U ∂t > 0

∂uv ∂y ↓

ωzωy ∂W

∂y ↑

ωiωi ↑ DT ↑ TKE ↓

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

24-28

PHYSICAL INTERPRETATION OF ωzωy∂ W/∂y

• ωzωy∂

W/∂y is key term leading to drag reduction

• ωzωy∂

W/∂y → ∂ W/∂y acts on ωzωy

• ωzωy ≈

∂u ∂y ∂u ∂z ∂u ∂y → upward eruption of near-wall low-speed fluid ∂u ∂z → lateral flanks of the low-speed streaks 200 400 600 800 200 400

x z

∂u ∂y ∂u ∂z located at the sides of high-speed streaks

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

25-28

MODELLING TURBULENT ENSTROPHY PRODUCTION

y z xn xs α ωyz

SIMPLIFIED TURBULENT ENSTROPHY EQUATION

1 2 ∂ ∂t

• ω2

y + ω2 z

• = ωzωyG −

∂ωy

∂y

2

−

∂ωz

∂y

2

Rotation of axis 1 2 ∂ω2

n

∂t = Snnω2

n −

∂ωn

∂y

2

Integration by Charpit’s method ωn = ωn,0 esin α cos αGt

• stretching

e−β2te−βy

• dissipation

, β = ∂ωn/∂t ∂ωn/∂y ∼ λy λt

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

26-28

OSCILLATION PERIOD VS. TERM 3

0.02 0.04 0.06 0.08 0.1 0.05 0.1 0.15 0.2 0.25 0.3 0.35

• ωzωy∂

W/∂y

g

R T=8 T=21 T=42 T=100

Drag reduction grows monotonically with global production term This happens up to optimum period

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

27-28

THANK YOU! REFERENCE

Ricco, P . Ottonelli, C. Hasegawa, Y. Quadrio, M. Changes in turbulent dissipation in a channel flow with oscillating walls

• J. Fluid Mech., 700, 77-104, 2012.

5 DECEMBER 2012

WALL-OSCILLATION DRAG-REDUCTION PROBLEM

28-28